(233.) An inflexible, straight bar, turning on an axis, is commonly called a lever. The arms of the lever are those parts of the bar which extend on each side of the axis.

The axis is called the fulcrum or prop.

(234.) Levers are commonly divided into three kinds, according to the relative positions of the power, the weight, and the fulcrum.

In a lever of the first kind, as in fig. 78., the fulcrum is between the power and weight.

In a lever of the second kind, as in fig. 79., the weight is between the fulcrum and power.

In a lever of the third kind, as in fig. 80., the power is between the fulcrum and weight.

(235.) In all these cases, the power will sustain the weight in equilibrium, provided its moment be equal to that of the weight. (184.) But the moment of the power is, in this case, equal to the product obtained by multiplying the power by its distance from the fulcrum; and the moment of the weight by multiplying the weight by its distance from the fulcrum. Thus, if the number of ounces in P, being multiplied by the number of inches in P F, be equal to the number of ounces in W, multiplied by the number of inches in W F, equilibrium will be established. It is evident from this, that as the distance of the power from the fulcrum increases in comparison to the distance of the weight from the fulcrum, in the same degree exactly will the proportion of the power to the weight diminish. In other words, the proportion of the power to the weight will be always the same as that of their distances from the fulcrum taken in a reverse order.

In cases where a small power is required to sustain or elevate a great weight, it will therefore be necessary either to remove the power to a great distance from the fulcrum, or to bring the weight very near it.

(236.) Numerous examples of levers of the first kind may be given. A crow-bar, applied to elevate a stone or other weight, is an instance. The fulcrum is another stone placed near that which is to be raised, and the power is the hand placed at the other end of the bar.

A handspike is a similar example.

A poker applied to raise fuel is a lever of the first kind, the fulcrum being the bar of the grate.

Scissors, shears, nippers, pincers, and other similar instruments are composed of two levers of the first kind; the fulcrum being the joint or pivot, and the weight the resistance of the substance to be cut or seized; the power being the fingers applied at the other end of the levers.

The brake of a pump is a lever of the first kind; the pump-rods and piston being the weight to be raised.

(237.) Examples of levers of the second kind, though not so frequent as those just mentioned, are not uncommon.

An oar is a lever of the second kind. The reaction of the water against the blade is the fulcrum. The boat is the weight, and the hand of the boatman the power.

The rudder of a ship or boat is an example of this kind of lever, and explained in a similar way.

The chipping knife is a lever of the second kind. The end attached to the bench is the fulcrum, and the weight the resistance of the substance to be cut, placed beneath it.

A door moved upon its hinges is another example.

Nut-crackers are two levers of the second kind; the hinge which unites them being the fulcrum, the resistance of the shell placed between them being the weight, and the hand applied to the extremity being the power.

A wheelbarrow is a lever of the second kind; the fulcrum being the point at which the wheel presses on the ground, and the weight being that of the barrow and its load, collected at their centre of gravity.

The same observation may be applied to all two-wheeled carriages, which are partly sustained by the animal which draws them.

(238.) In a lever of the third kind, the weight, being more distant from the fulcrum than the power, must be proportionably less than it. In this instrument, therefore, the power acts upon the weight to a mechanical disadvantage, inasmuch as a greater power is necessary to support or move the weight than would be required if the power were immediately applied to the weight, without the intervention of a machine. We shall, however, hereafter show that the advantage which is lost in force is gained in despatch, and that in proportion as the weight is less than the power which moves it, so will the speed of its motion be greater than that of the power.

Hence a lever of the third kind is only used in cases where the exertion of great power is a consideration subordinate to those of rapidity and despatch.

The most striking example of levers of the third kind is found in the animal economy. The limbs of animals are generally levers of this description. The socket of the bone is the fulcrum; a strong muscle attached to the bone near the socket is the power; and the weight of the limb, together with whatever resistance is opposed to its motion, is the weight. A slight contraction of the muscle in this case gives a considerable motion to the limb: this effect is particularly conspicuous in the motion of the arms and legs in the human body; a very inconsiderable contraction of the muscles at the shoulders and hips giving the sweep to the limbs from which the body derives so much activity.

The treddle of the turning lathe is a lever of the third kind. The hinge which attaches it to the floor is the fulcrum, the foot applied to it near the hinge is the power, and the crank upon the axis of the fly-wheel, with which its extremity is connected, is the weight.

Tongs are levers of this kind, as also the shears used in shearing sheep. In these cases the power is the hand placed immediately below the fulcrum or point where the two levers are connected.

(239.) When the power is said to support the weight by means of a lever or any other machine, it is only meant that the power keeps the machine in equilibrium, and thereby enables it to sustain the weight. It is necessary to attend to this distinction, to remove the difficulty which may arise from the paradox of a small power sustaining a great weight.

In a lever of the first kind, the fulcrum F, fig. 78., or axis, sustains the united forces of the power and weight.

In a lever of the second kind, if the power be supposed to act over a wheel R, fig. 79., the fulcrum F sustains a pressure equal to the difference between the power and weight, and the axis of the wheel R sustains a pressure equal to twice the power; so that the total pressures on F and R are equivalent to the united forces of the power and weight.

In a lever of the third kind similar observations are applicable. The wheel R, fig. 80., sustains a pressure equal to twice the power, and the fulcrum F sustains a pressure equal to the difference between the power and weight.

These facts may be experimentally established by attaching a string to the lever immediately over the fulcrum, and suspending the lever by that string from the arm of a balance. The counterpoising weight, when the fulcrum is removed, will, in the first case, be equal to the sum of the weight and power, and in the last two cases equal to their difference.

(240.) We have hitherto omitted the consideration of the effect of the weight of the lever itself. If the centre of gravity of the lever be in the vertical line through the axis, the weight of the instrument will have no other effect than to increase the pressure on the axis by its own amount. But if the centre of gravity be on the same side of the axis with the weight, as at G, it will oppose the effect of the power, a certain part of which must therefore be allowed to support it. To ascertain what part of the power is thus expended, it is to be considered that the moment of the weight of the lever collected at G, is found by multiplying that weight by the distance G F. The moment of that part of the power which supports this must be equal to it; therefore, it is only necessary to find how much of the power multiplied by P F will be equal to the weight of the lever multiplied by G F. This is a question in common arithmetic.

If the centre of gravity of the lever be at a different side of the axis from the weight, as at G′, the weight of the instrument will co-operate with the power in sustaining the weight W. To determine what portion of the weight W is thus sustained by the weight of the lever, it is only necessary to find how much of W, multiplied by the distance W F, is equal to the weight of the lever multiplied by G′ F.

In these cases the pressure on the fulcrum, as already estimated, will always be increased by the weight of the lever.

(241.) The sense in which a small power is said to sustain a great weight, and the manner of accomplishing this, being explained, we shall now consider how the power is applied in moving the weight. Let P W, fig. 81., be the places of the power and weight, and F that of the fulcrum, and let the power be depressed to P′ while the weight is raised to W′. The space P P′ evidently bears the same proportion to W W′, as the arm P F to W F. Thus if P F be ten times W F, P P′ will be ten times W W′. A power of one pound at P being moved from P to P′, will carry a weight of ten pounds from W to W′. But in this case it ought not to be said, that a lesser weight moves a greater, for it is not difficult to show, that the total expenditure of force in the motion of one pound from P to P′ is exactly the same as in the motion of ten pounds from W to W′. If the space P P′ be ten inches, the space W W′ will be one inch. A weight of one pound is therefore moved through ten successive inches, and in each inch the force expended is that which would be sufficient to move one pound through one inch. The total expenditure of force from P to P′ is ten times the force necessary to move one pound through one inch, or what is the same, it is that which would be necessary to move ten pounds through one inch. But this is exactly what is accomplished by the opposite end W of the lever; for the weight W is ten pounds, and the space W W′ is one inch.

If the weight W of ten pounds could be conveniently divided into ten equal parts of one pound each, each part might be separately raised through one inch, without the intervention of the lever or any other machine. In this case, the same quantity of power would be expended, and expended in the same manner as in the case just mentioned.

It is evident, therefore, that when a machine is applied to raise a weight or to overcome resistance, as much force must be really used as if the power were immediately applied to the weight or resistance. All that is accomplished by the machine is to enable the power to do that by a succession of distinct efforts which should be otherwise performed by a single effort. These observations will be found to be applicable to all machines whatever.

(242.) Weighing machines of almost every kind, whether used for commercial or philosophical purposes, are varieties of the lever. The common balance, which, of all weighing machines, is the most perfect and best adapted for ordinary use, whether in commerce or experimental philosophy, is a lever with equal arms. In the steel-yard one weight serves as a counterpoise and measure of others of different amount, by receiving a leverage variable according to the varying amount of the weight against which it acts. A detailed account of such instruments will be found in Chapter XXI.

(243.) We have hitherto considered the power and weight as acting on the lever, in directions perpendicular to its length and parallel to each other. This does not always happen. Let A B, fig. 83., be a lever whose fulcrum is F, and let A R be the direction of the power, and B S the direction of the weight. If the lines R A and S B be continued, and perpendiculars F C and F D drawn from the fulcrum to those lines, the moment of the power will be found by multiplying the power by the line F C, and the moment of the weight by multiplying the weight by F D. If these moments be equal, the power will sustain the weight in equilibrium. (185).

It is evident, that the same reasoning will be applicable when the arms of the lever are not in the same direction. These arms may be of any figure or shape, and may be placed relatively to each other in any position.

(244.) In the rectangular lever the arms are perpendicular to each other, and the fulcrum F, fig. 84., is at the right angle. The moment of the power, in this case, is P multiplied by A F, and that of the weight W multiplied by B F. When the instrument is in equilibrium these moments must be equal.

When the hammer is used for drawing a nail, it is a lever of this kind: the claw of the hammer is the shorter arm; the resistance of the nail is the weight; and the hand applied to the handle the power.

(245.) When a beam rests on two props A B, fig. 85., and supports, at some intermediate place C, a weight W, this weight is distributed between the props in a manner which may be determined by the principles already explained. If the pressure on the prop B be considered as a power sustaining the weight W, by means of the lever of the second kind B A, then this power multiplied by B A must be equal to the weight multiplied by C A. Hence the pressure on B will be the same fraction of the weight as the part A C is of A B. In the same manner it may be proved, that the pressure on A is the same fraction of the weight as B C is of B A. Thus, if A C be one third, and therefore B C two thirds of B A, the pressure on B will be one third of the weight, and the pressure on A two thirds of the weight.

It follows from this reasoning, that if the weight be in the middle, equally distant from B and A, each prop will sustain half the weight. The effect of the weight of the beam itself may be determined by considering it to be collected at its centre of gravity. If this point, therefore, be equally distant from the props, the weight of the beam will be equally distributed between them.

According to these principles, the manner in which a load borne on poles between two bearers is distributed between them may be ascertained. As the efforts of the bearers and the direction of the weight are always parallel; the position of the poles relatively to the horizon makes no difference in the distribution of the weights between the bearers. Whether they ascend or descend, or move on a level plane, the weight will be similarly shared between them.

If the beam extend beyond the prop, as in fig. 86., and the weight be suspended at a point not placed between them, the props must be applied at different sides of the beam. The pressures which they sustain may be calculated in the same manner as in the former case. The pressure of the prop B may be considered as a power sustaining the weight W by means of the lever B C. Hence, the pressure of B, multiplied by B A, must be equal to the weight W multiplied by A C. Therefore, the pressure on B bears the same proportion to the weight as A C does to A B. In the same manner, considering B as a fulcrum, and the pressure of the prop A as the power, it may be proved that the pressure of A bears the same proportion to the weight as the line B C does to A B. It therefore appears, that the pressure on the prop A is greater than the weight.

(246.) When great power is required, and it is inconvenient to construct a long lever, a combination of levers may be used. In fig. 87. such a system of levers is represented, consisting of three levers of the first kind. The manner in which the effect of the power is transmitted to the weight may be investigated by considering the effect of each lever successively. The power at P produces an upward force at P′, which bears to P the same proportion as P′ F to P F. Therefore, the effect at P′ is as many times the power as the line P F is of P′ F. Thus, if P F be ten times P′ F, the upward force at P′ is ten times the power. The arm P′ F′ of the second lever is pressed upwards by a force equal to ten times the power at P. In the same manner this may be shown to produce an effect at P″ as many times greater than P′ as P′ F′ is greater than P″ F′. Thus, if P′ F′ be twelve times P″ F′, the effect at P″ will be twelve times that of P′. But this last was ten times the power, and therefore the P″ will be one hundred and twenty times the power. In the same manner it may be shown that the weight is as many times greater than the effect at P″ as P″ F″ is greater than W F″. If P″ F″ be five times W F″, the weight will be five times the effect at P″. But this effect is one hundred and twenty times the power, and therefore the weight would be six hundred times the power.

In the same manner the effect of any compound system of levers may be ascertained by taking the proportion of the weight to the power in each lever separately, and multiplying these numbers together. In the example given, these proportions are 10, 12, and 5, which multiplied together give 600. In fig. 87. the levers composing the system are of the first kind; but the principles of the calculation will not be altered if they be of the second or third kind, or some of one kind and some of another.

(247.) That number which expresses the proportion of the weight to the equilibrating power in any machine, we shall call the power of the machine. Thus, if, in a lever, a power of one pound support a weight of ten pounds, the power of the machine is ten. If a power of 2lbs. support a weight of 11lbs., the power of the machine is 51/2, 2 being contained in 11 51/2 times.

(248.) As the distances of the power and weight from the fulcrum of a lever may be varied at pleasure, and any assigned proportion given to them, a lever may always be conceived having a power equal to that of any given machine. Such a lever may be called, in relation to that machine, the equivalent lever.

As every complex machine consists of a number of simple machines acting one upon another, and as each simple machine may be represented by an equivalent lever, the complex machine will be represented by a compound system of equivalent levers. From what has been proved in (246.), it therefore follows that the power of a complex machine may be calculated by multiplying together the powers of the several simple machines of which it is composed.